OFFSET
0,4
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
EXAMPLE
a(3) = 3 because there are 2 distinct orientations of the triangle K_3 plus the empty graph on 3 vertices.
PROG
(PARI) \\ See links in A339645 for combinatorial species functions.
oedges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
ographsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 3^oedges(p) * sMonomial(p)); s/n!}
ographs(n)={sum(k=0, n, ographsCycleIndex(k)*x^k) + O(x*x^n)}
trees(n, k)={sRevert(x*sv(1)/sExp(k*x*sv(1) + O(x^n)))}
cycleIndexSeries(n)={my(g=ographs(n), tr=trees(n, 2), tu=tr-tr^2); sSolve( g/sExp(tu), tr )*symGroupSeries(n)}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 27 2020
(PARI) \\ faster stand-alone version
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, (v[i]-1)\2)}
seq(n)={Vec(sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 3^edges(p) * prod(i=1, #p, my(d=p[i]); (1-x^d)^2 + O(x*x^(n-k))) ); x^k*s/k!)/(1-x^2))} \\ Andrew Howroyd, Jan 22 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Goran Kilibarda, Vladeta Jovovic, Jan 14 2005
EXTENSIONS
a(0)=1 prepended and terms a(9) and beyond from Andrew Howroyd, Dec 27 2020
STATUS
approved